L. Ingber

English

Research Papers in Economics

%A L. Ingber
%T Path-integral evolution of multivariate systems with moderate noise
%J Phys. Rev. E
%V 51
%N 2
%P 1616-1619
%D 1995
Path-integral evolution of multivariate systems with moderate noise
Lester Ingber
Lester Ingber Research, P.O. Box 857, McLean, VA 22101
ingber@alumni.caltech.edu
A non Monte Carlo path-integral algorithm that is particularly adept at handling nonlinear
Lagrangians is extended to multivariate systems. This algorithm is particularly accurate for systems with
moderate noise.
PA CS Nos.: 02.70.Rw, 05.40.+j, 02.50.-rPath-integral evolution of multivariate systems -2-  Lester Ingber
PATHINT is a non Monte Carlo histogram C computer code developed to evolve an n-dimensional
system (subject to machine constraints) based on a generalization of an algorithm demonstrated by
Wehner and Wolfer to be extremely robust for nonlinear Lagrangians with moderate noise [1-3].
PATHINT was recently used in a neuroscience study [4], where it was observed how difﬁcult it can be
ev  en in two dimensions to get good resolution because of CPU constraints on Sun SPARC 2 and 10MP
machines.  Here it appears that the resolution is quite satisfactory on these machines.
The system selected for this paper to illustrate the use of PATHINT is the classical analog of a
quantum system. “Quantum chaos” was a term used to describe the observation of chaos in the classical
trajectories of the Hamilton’s equations of motion,
˙ p =−
∂ H
∂ q
,
˙ q =
∂ H
∂ p
, (1)
where a Hamiltonian of the form
H =
1
2
p2 + ˆ Φ (q),
ˆ Φ=
1
2
(∇Φ )2 −
1
2
D∇ 2Φ , (2)
with a coefﬁcient D was considered in a q dimension ≥ 2, representing double this dimension in the
corresponding phase space of q and p [5].
In ref. [5] a potential Φ was considered,
Φ= 2x4 +
3
5
y4 + ε xy(x − y)2 , (3)
in a classical Fokker-Planck system
∂ Pt =∇ .(P∇Φ ) +
D
2
∇ 2P . (4)
A slight change of notation is used in this paper. They posed the question as to just what properties suchPath-integral evolution of multivariate systems -3-  Lester Ingber
classical systems might possess?
This paper does not at all deal with the quantum system described above, but it does deal in detail
with the associated classical system, and it does give some answers to the above posed question. This
study should be considered as illustrating a particular numerical approach that promises to be quite useful
in studying the evolution of such classical systems. The results obtained can be considered as
“experimental” data on the exact region of such classical transformations. The results here are negative
with respect to any unusual or interesting activity in the parameter region observed in the quantum
mechanical calculation. This should at least help other investigators who might tend to focus on this
region in the classical system, based on the results obtained for the associated quantum system.
This paper computes the path integral of the classical system in terms of its Lagrangian L.
P[qt|qt0]dq(t) = ∫
...∫ Dqexp



− min
t
t0 ∫ dt′ L



δ ( (q(t0) = q0) )δ ( (q(t) = qt) )  ,
Dq =
u→ ∞
lim
u+1
ρ =1 Π g1/2
i Π (2π ∆ t)− 1/2dqi
ρ ,
L(˙ qi,qi,t) =
1
2
(˙ qi − gi)gii′ (˙ qi′ − gi′ ),
gii′ = (gii′ )− 1 ,
g = det(gii′ ).  (5)
Here the diagonal diffusion terms are gxx = gyy = D and the drift terms are gi = −∂Φ /∂ qi. If the
diffusions terms are not constant, then there are additional terms [6].
The histogram procedure recognizes that the distribution can be numerically approximated to a high
degree of accuracy by sums of rectangles of height Pi and width ∆ qi at points qi. For convenience, just
consider a one-dimensional system. The above path-integral representation can be rewritten, for each of
its intermediate integrals, asPath-integral evolution of multivariate systems -4-  Lester Ingber
P(x;t +∆ t) =∫ dx′ [g1/2(2π ∆ t)− 1/2 exp(− L ∆ t)]P(x′ ;t)
=∫ dx′ G(x, x′ ;∆ t)P(x′ ;t),
P(x;t) =
N
i=1 Σ π (x − xi)Pi(t),
π (x − xi) =





1, (xi −
1
2
∆ xi− 1) ≤ x ≤ (xi +
1
2
∆ xi),
0, otherwise  .
(6)
This yields
Pi(t +∆ t) = Tij(∆ t)P j(t),
Tij(∆ t) =
2
∆ xi− 1 +∆ xi ∫
xi+∆ xi/2
xi−∆ xi− 1/2
dx∫
x j+∆ x j/2
x j−∆ x j− 1/2
dx′ G(x, x′ ;∆ t). ( 7 )
Tij is a banded matrix representing the Gaussian nature of the short-time probability centered about the
(possibly time-dependent) drift.
This histogram procedure was extended to two dimensions using a matrix Tijkl [7]. Explicit
dependence of L on time t also can be included without complications. Care must be used in developing
the mesh in ∆ qi, which is strongly dependent on the diagonal elements of the diffusion matrix, e.g.,
∆ qi ≈ (∆ tgii)1/2 . (8)
This constrains the dependence of the covariance of each variable to be a (nonlinear) function of that
variable in order to present a straightforward rectangular underlying mesh.
Since integration is inherently a smoothing process [8], ﬁtting the data with the short-time
probability distribution, effectively using an integral over this epoch, permits the use of coarser meshes
than the corresponding stochastic differential equation. For example, the coarser resolution is
appropriate, as typically required, for a numerical solution of the time-dependent path integral.  ByPath-integral evolution of multivariate systems -5-  Lester Ingber
considering the contributions to the ﬁrst and second moments, conditions on the time and variable meshes
can be derived [1].  The time slice essentially is determined by ∆ t ≤ L
− 1,w here L is the uniform
Lagrangian, respecting ranges giving the most important contributions to the probability distribution P.
Thus ∆ t is roughly measured by the diffusion divided by the square of the drift.
Such calculations are useful in many disciplines, e.g., some ﬁnancial instruments [8,9].  Monte
Carlo algorithms for path integrals are well known to have extreme difﬁculty in evolving nonlinear
systems with multiple optima [10], but this algorithm does very well on such systems. The PATHINT
code was tested against the test problems given in previous one-dimensional systems [1,2], and it was
established that the method of images for both Dirichlet and Neumann boundary conditions is as accurate
as the boundary element methods for the systems investigated.  Two-dimensional runs were tested by
using cross products of one-dimensional examples whose analytic solutions are known.
Attempts were made to process the same system considered for the quantum case [5].  Therefore,
the diffusion was taken to be D = 0.  2. Since they selected an harmonic oscillator basis for their
eigenvalue study, is was assumed that natural boundary conditions are appropriate for this study, and
ranges of x and y were tested to ensure that this was reasonable. A band of three units on each side of the
short-time distribution was sufﬁcient for these runs. For ε ≤ 5, the range of x was taken to be ±3 and the
range of y was taken to be ±6.
A mesh of ∆ t = 0. 1 was reasonable to calculate the evolution of this system for 0.1 ≤ ε ≤ 0. 5. The
quantum study observed chaos at ε ≥ 0. 14, but the classical system appears to be very stable with a single
peak in its probability density up through ε = 0.  5. The time mesh was tested by performing several
calculations at time meshes of ∆ t = 0. 01 on a Sun SPARC 10MP. All other calculations reported here
were performed on a Sun SPARC 2.
Figures 1(a) and 1(b) show the evolution of the distribution for ε = 0.1, t = 10 and 100, i.e., after
100 and 1000 foldings of the path integral, respectively. The distribution starts at single peaks at the
imposed initial condition, a δ function at the origin, i.e., x =− 0. 0302 and y =− 0. 0603 with this mesh,
and swell out their stable structures within t < 0.03.  Note the stability over the duration of the
calculation.  A similar stability was noted for ε < 0.55, as illustrated in Figures 2(a) and 2(b) for ε = 0.3
and 0.5 after 1000 foldings.Path-integral evolution of multivariate systems -6-  Lester Ingber
For values of ε > 0.5, a mesh of ∆ t = 0. 05 was used to investigate the onset of instabilities which
were noted for higher ε with coarser meshes. To two signiﬁcant ﬁgures, this ﬁrst occurs at ε = 0. 55. As
ε increases, so does the peak spread quite early along a diagonal in the evolving distribution.  As the time
of the calculation increased, there was concern that the boundaries were being approached. Therefore, for
these runs, the range of x was increased to ±5 and the range of y was increased to ±10. Figure 3
illustrates the evolution at time t = 5. 0 (100 foldings) and t = 100 (2000 foldings) with ε = 0.55. Figure 4
illustrates the early structure for ε = 0.6 at t = 15 after 300 foldings of ∆ t = 0. 05.
The PATHINT algorithm utilized here will be used to explore the time evolution of other Fokker-
Planck systems in the presence of moderate noise. As mentioned above, such problems arise in many
systems ranging from neocortical interactions to ﬁnancial markets.  Also, we can now accurately examine
long-time correlations of chaotic models as multiplicative noise is increased to moderate and strong
levels; many chaotic models do not include such levels of noise as is found in the systems they are
attempting to model. A project is now underway under an award of Cray computer time from the
Pittsburgh Supercomputing Center through the National Science Foundation (NSF), the Parallelizing ASA
and PATHINT Project (PAPP) further developing this code for large systems [11].Path-integral evolution of multivariate systems -7-  Lester Ingber
FIGURE CAPTIONS
FIG. 1. Probability density for ε = 0. 1, for ∆ t = 0. 1 (a) after 100 foldings (t = 10) and (b) after
1000 foldings (t = 100).
FIG. 2. Probability density after 1000 foldings of ∆ t = 0. 1 (t = 100) for ε set to (a) 0.3 and (b) 0.5.
FIG. 3. Probability density for ε set to 0.55 using ∆ t = 0. 05, (a) at t = 5 after 100 foldings, and (b)
at t = 100 after 2000 foldings. As discussed in the text, the ranges of x and y for ε > 0.5 were increased.
FIG. 4. Probability density at t = 15, after 300 foldings of ∆ t = 0. 05, for ε = 0.6.Path-integral evolution of multivariate systems -8-  Lester Ingber
REFERENCES
[1]  M.F. Wehner and W.G. Wolfer, ‘‘Numerical evaluation of path-integral solutions to Fokker-Planck
equations. I.,’’ Phys. Rev. A 27, 2663-2670 (1983).
[2]  M.F. Wehner and W.G. Wolfer, ‘‘Numerical evaluation of path-integral solutions to Fokker-Planck
equations. II. Restricted stochastic processes,’’ Phys. Rev. A 28, 3003-3011 (1983).
[3]  M.F. Wehner and W.G. Wolfer, ‘‘Numerical evaluation of path integral solutions to Fokker-Planck
equations. III. Time and functionally dependent coefﬁcients,’’ Phys. Rev. A 35, 1795-1801 (1987).
[4]  L. Ingber, ‘‘Statistical mechanics of neocortical interactions: Path-integral evolution of short-term
memory,’’ Phys. Rev. E 49, 4652-4664 (1994).
[5]  M.M. Millonas and L.E. Reichl, ‘‘Stochastic chaos in a class of Fokker-Planck equations,’’ Phys.
Rev. Lett. 68, 3125-3128 (1992).
[6]  F. Langouche, D. Roekaerts, and E. Tirapegui, Functional Integration and Semiclassical
Expansions (Reidel, Dordrecht, The Netherlands, 1982).
[7]  L. Ingber, H. Fujio, and M.F. Wehner, ‘‘Mathematical comparison of combat computer models to
exercise data,’’ Mathl. Comput. Modelling 15, 65-90 (1991).
[8]  L. Ingber, ‘‘Statistical mechanical aids to calculating term structure models,’’ Phys. Rev. A 42,
7057-7064 (1990).
[9]  L. Ingber, M.F. Wehner, G.M. Jabbour, and T.M. Barnhill, ‘‘Application of statistical mechanics
methodology to term-structure bond-pricing models,’’ Mathl. Comput. Modelling 15, 77-98 (1991).
[10]  K. Binder and D. Stauffer, ‘‘A simple introduction to Monte Carlo simulations and some specialized
topics,’’ in Applications of the Monte Carlo Method in Statistical Physics, ed. by K. Binder
(Springer-Verlag, Berlin, 1985), p. 1-36.
[11]Path-integral evolution of multivariate systems Lester Ingber
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Figure 4
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P

A simple introduction to Monte Carlo simulations and some specialized topics,’’

Application of statistical mechanics methodology to term-structure bond-pricing models,’’

Functional Integration and Semiclassical Expansions (Reidel,

Mathematical comparison of combat computer models to exercise data,’’

Numerical evaluation of path integral solutions to Fokker-Planck equations. III. Time and functionally dependent coefﬁcients,’’

Numerical evaluation of path-integral solutions to Fokker-Planck equations.

Numerical evaluation of path-integral solutions to Fokker-Planck equations. II. Restricted stochastic processes,’’

Path-integral evolution of multivariate systems Lester Ingber

Statistical mechanical aids to calculating term structure models,’’

Statistical mechanics of neocortical interactions: Path-integral evolution of short-term memory,’’

Stochastic chaos in a class of Fokker-Planck equations,’’

Path-integral evolution of multivariate systems with moderate noise

Path-integral evolution of multivariate systems with moderate noise

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Research Papers in Economics